Decaying barotropic QG beta-plane turbulence

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An example of decaying barotropic quasi-geostrophic turbulence on a beta plane.

using FourierFlows, Plots, Printf, Random

using Statistics: mean
using FFTW: irfft

import GeophysicalFlows.BarotropicQG
import GeophysicalFlows.BarotropicQG: energy, enstrophy

Choosing a device: CPU or GPU

dev = CPU()     # Device (CPU/GPU)

Numerical parameters and time-stepping parameters

      n = 128            # 2D resolution = n²
stepper = "FilteredRK4"  # timestepper
     dt = 0.05           # timestep
 nsteps = 2000           # total number of time-steps
 nsubs  = 10             # number of time-steps for intermediate logging/plotting (nsteps must be multiple of nsubs)

Physical parameters

L = 2π        # domain size
β = 10.0      # planetary PV gradient
μ = 0.0       # bottom drag

Problem setup

We initialize a Problem by providing a set of keyword arguments. Not providing a viscosity coefficient ν leads to the module's default value: ν=0. In this example numerical instability due to accumulation of enstrophy at high wavenumbers is taken care with the FilteredTimestepper we picked.

prob = BarotropicQG.Problem(dev; nx=n, Lx=L, β=β, μ=μ, dt=dt, stepper=stepper)

and define some shortcuts

sol, clock, vars, params, grid = prob.sol, prob.clock, prob.vars, prob.params, prob.grid
x, y = grid.x, grid.y

Setting initial conditions

Our initial condition consist of a flow that has power only at wavenumbers with $8 < \frac{L}{2\pi} \sqrt{k_x^2 + k_y^2} < 10$ and initial energy $E_0$:

E₀ = 0.1 # energy of initial condition

K = @. sqrt(grid.Krsq)                          # a 2D array with the total wavenumber
k = [grid.kr[i] for i=1:grid.nkr, j=1:grid.nl]  # a 2D array with the zonal wavenumber

Random.seed!(1234)
qih = randn(Complex{eltype(grid)}, size(sol))
@. qih = ifelse(K < 2  * 2π/L, 0, qih)
@. qih = ifelse(K > 10 * 2π/L, 0, qih)
@. qih = ifelse(k == 0 * 2π/L, 0, qih)   # no power at zonal wavenumber k=0 component
Ein = energy(qih, grid)           # compute energy of qi
qih *= sqrt(E₀ / Ein)             # normalize qi to have energy E₀
qi = irfft(qih, grid.nx)

BarotropicQG.set_zeta!(prob, qi)

Let's plot the initial vorticity field:

p1 = heatmap(x, y, vars.q',
         aspectratio = 1,
              c = :balance,
           clim = (-12, 12),
          xlims = (-grid.Lx/2, grid.Lx/2),
          ylims = (-grid.Ly/2, grid.Ly/2),
         xticks = -3:3,
         yticks = -3:3,
         xlabel = "x",
         ylabel = "y",
          title = "initial vorticity ζ=∂v/∂x-∂u/∂y",
     framestyle = :box)

p2 = contourf(x, y, vars.psi',
        aspectratio = 1,
             c = :viridis,
        levels = range(-0.65, stop=0.65, length=10),
          clim = (-0.65, 0.65),
         xlims = (-grid.Lx/2, grid.Lx/2),
         ylims = (-grid.Ly/2, grid.Ly/2),
        xticks = -3:3,
        yticks = -3:3,
        xlabel = "x",
        ylabel = "y",
         title = "initial streamfunction ψ",
    framestyle = :box)

l = @layout Plots.grid(1, 2)
p = plot(p1, p2, layout=l, size=(900, 400))

Diagnostics

Create Diagnostics – energy and enstrophy functions are imported at the top.

E = Diagnostic(energy, prob; nsteps=nsteps)
Z = Diagnostic(enstrophy, prob; nsteps=nsteps)
diags = [E, Z] # A list of Diagnostics types passed to "stepforward!" will  be updated every timestep.

Output

We choose folder for outputing .jld2 files and snapshots (.png files).

filepath = "."
plotpath = "./plots_decayingbetaturb"
plotname = "snapshots"
filename = joinpath(filepath, "decayingbetaturb.jld2")

Do some basic file management,

if isfile(filename); rm(filename); end
if !isdir(plotpath); mkdir(plotpath); end

and then create Output.

get_sol(prob) = sol # extracts the Fourier-transformed solution
get_u(prob) = irfft(im * grid.l .* grid.invKrsq .* sol, grid.nx)
out = Output(prob, filename, (:sol, get_sol), (:u, get_u))

Visualizing the simulation

We define a function that plots the vorticity and streamfunction and their corresponding zonal mean structure.

function plot_output(prob)
  ζ = prob.vars.zeta
  ψ = prob.vars.psi
  ζ̄ = mean(ζ, dims=1)'
  ū = mean(prob.vars.u, dims=1)'

  pζ = heatmap(x, y, ζ',
       aspectratio = 1,
            legend = false,
                 c = :balance,
              clim = (-12, 12),
             xlims = (-grid.Lx/2, grid.Lx/2),
             ylims = (-grid.Ly/2, grid.Ly/2),
            xticks = -3:3,
            yticks = -3:3,
            xlabel = "x",
            ylabel = "y",
             title = "vorticity ζ=∂v/∂x-∂u/∂y",
        framestyle = :box)

  pψ = contourf(x, y, ψ',
       aspectratio = 1,
            legend = false,
                 c = :viridis,
            levels = range(-0.65, stop=0.65, length=10),
              clim = (-0.65, 0.65),
             xlims = (-grid.Lx/2, grid.Lx/2),
             ylims = (-grid.Ly/2, grid.Ly/2),
            xticks = -3:3,
            yticks = -3:3,
            xlabel = "x",
            ylabel = "y",
             title = "streamfunction ψ",
        framestyle = :box)

  pζm = plot(ζ̄, y,
            legend = false,
         linewidth = 2,
             alpha = 0.7,
            yticks = -3:3,
             xlims = (-2.2, 2.2),
            xlabel = "zonal mean ζ",
            ylabel = "y")
  plot!(pζm, 0*y, y, linestyle=:dash, linecolor=:black)

  pum = plot(ū, y,
            legend = false,
         linewidth = 2,
             alpha = 0.7,
            yticks = -3:3,
             xlims = (-0.55, 0.55),
            xlabel = "zonal mean u",
            ylabel = "y")
  plot!(pum, 0*y, y, linestyle=:dash, linecolor=:black)

  l = @layout Plots.grid(2, 2)
  p = plot(pζ, pζm, pψ, pum, layout = l, size = (900, 800))

  return p
end

Time-stepping the Problem forward

We time-step the Problem forward in time.

startwalltime = time()

p = plot_output(prob)

anim = @animate for j = 0:round(Int, nsteps/nsubs)

  if j % (1000 / nsubs) == 0
    cfl = clock.dt * maximum([maximum(vars.u) / grid.dx, maximum(vars.v) / grid.dy])

    log = @sprintf("step: %04d, t: %d, cfl: %.2f, E: %.4f, Q: %.4f, walltime: %.2f min",
      clock.step, clock.t, cfl, E.data[E.i], Z.data[Z.i], (time()-startwalltime)/60)

    println(log)
  end

  p[1][1][:z] = vars.zeta
  p[1][:title] = "vorticity, t="*@sprintf("%.2f", clock.t)
  p[3][1][:z] = vars.psi
  p[2][1][:x] = mean(vars.zeta, dims=1)'
  p[4][1][:x] = mean(vars.u, dims=1)'

  stepforward!(prob, diags, nsubs)
  BarotropicQG.updatevars!(prob)

end

mp4(anim, "barotropicqg_betadecay.mp4", fps=12)

Save

Finally save the last snapshot.

savename = @sprintf("%s_%09d.png", joinpath(plotpath, plotname), clock.step)
savefig(savename)

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